( P) det. constitute the cofactor matrix, or the matrix of the cofactors: com P = c. ( 1) det
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1 Aendix C Tranfer Matrix Inverion To invert one matrix P, the variou te are a follow: calculate it erminant ( P calculate the cofactor ij of each element, tarting from the erminant of the correonding minor matrix P { ij } of P, i.e. the matrix P, in which one remove the row i and the column j : ( i+ j ( { ij} contitute the cofactor matrix, or the matrix of the cofactor: c P (C. ij ( calculate the tranoe cofactor matrix com (P T, adjugate of P calculate the invere of P by: com P c ij (C. com( P P T ( P (C. Thee algebraic calculation are heavy, rather than comlex. We will, however, carry them out for the IPM-SM, for inverting the tranfer matrix P (equation (B.7; thi reult i indeed neceary to calculate the control vector. The following calculation will not ail the intermediate calculation of the coefficient c ij of the adjugate of the P matrix, ince it i enough to: Direct Eigen Control for Induction Machine and Synchronou Motor, Firt Edition. Jean Claude Alacoque. 0 John Wiley & Son, td. Publihed 0 by John Wiley & Son, td.
2 Aendix C calculate the erminant of the minor matrice of P, uing Cramer rule give to the erminant, the correonding ign ( i + j invert the ubcrit i and j to create the coefficient c ij, of the adjugate matrix: c c (C. ij To calculate the invere matrix P, we will thu calculate firtly the erminant of the tranfer matrix: ( P. If ij give the invere matrix coefficient, we then will calculate each coefficient of the adjugate matrix: The invere matrix coefficient of P. ( P ij cij cji C. Tranfer Matrix Determinant Calculation ji (C.5 ij will be derived by the diviion of c ij by the erminant While develoing ( P from it firt column, the tranfer matrix erminant i calculated eaily: ( P ( μ μ ( μ μ ( μ μ ( μ μ ( μ μ q ( q μ ( μ + mq mq (C.6 The lat factor of the numerator can be written by uing equation (., the olution of the eigenvalue equation, alied to the eigenvalue μ : ( ( μ μ (C.7 mq q Thi exreion make it oible to factorize ( μ in the exreion of the erminant and in variou term of the invere matrix which we will then calculate. ( P ( μ q (C.8 mq ( μ μ ( μ μ ( μ μ ( μ μ ( μ μ ( μ μ We can now calculate each coefficient of P, the tranfer matrix invere. C. Firt Row, Firt Column ( P ( μ + q R ( μ μ ( μ μ ( μ μ ( μ μ ( μ μ mq (C.9
3 Aendix C 5 q ( μ μ (C.0 C. Firt Row, Second Column q ( P ( μ μ ( μ μ ( μ μ ( μ μ ( μ μ μ + q R q ( μ μ (C. (C. C. Firt Row, Third Column ( P ( μ μ ( μ μ ( μ μ q ( ( μ + μ μ μμ μ mq q mq (C. We can notice here that μ + μ 0 and μ μ ; moreover, equation ( C.7, make it oible to reduce: ( μ ( μ μ ( μ μ ( μ μ + q R ( P mq μ ( μ + q q R ( μ μ q q q ( μ μ ( μ μ ( μ μ (C. (C.5 C.5 Firt Row, Fourth Column ( P ( μ + + μ q R q ( μ ( μ μ ( μ μ ( μ μ + q R mq ( μ + μ q q q ( μ μ ( μ μ ( μ μ (C.6 (C.7
4 6 Aendix C C.6 Second Row, Firt Column q ( P ( μ μ ( μ μ ( μ μ ( μ μ ( μ μ (C.8 μ + q R q ( μ μ (C.9 C.7 Second Row, Second Column ( P ( μ + q q R ( μ μ ( μ μ ( μ μ ( μ μ ( μ μ ( q μ ( q μ ( μ μ q (C.0 (C. By uing equation (. and (., one can reduce thi to: mq ( μ μ (C. C.8 Second Row, Third Column ( ( μ μ ( μ μ ( μ μ q P ( μμ μ μ q q ( μ q q μμ μ( q μ μ μ μ μ μ μ q ( ( ( Thi new exreion can be now written uing equation (. and (.: ( ( μ μ ( μ μ ( μ μ μ + μ mq q q (C. (C. (C.5
5 Aendix C 7 C.9 Second Row, Fourth Column ( P ( μ μ ( μ μ ( μ μ q ( μ + μ q q ( q R q ( q R μ + μ + μ + μ μ μ μ μ μ q ( ( ( Thi new exreion can be written, again uing equation (. and (.: ( R ( μ μ ( μ μ ( μ μ μ μ + mq q q (C.6 (C.7 (C.8 C.0 Third Row, Firt Column 0 (C.9 C. Third Row, Second Column 0 (C.0 C. Third Row, Third Column ( P ( μ ( μ μ ( μ μ ( μ μ μ + q R mq μ ( μ μ ( μ μ ( μ μ q (C. (C. C. Third Row, Fourth Column ( P ( μ ( μ μ ( μ μ ( μ μ + q R mq (C.
6 8 Aendix C ( μ μ ( μ μ ( μ μ q (C. C. Fourth Row, Firt Column 0 (C.5 C.5 Fourth Row, Second Column 0 (C.6 C.6 Fourth Row, Third Column ( P ( μ μ ( μ μ ( μ μ ( μ μ + q R mq (C.7 q μ ( μ μ ( μ μ ( μ μ (C.8 C.7 Fourth Row, Fourth Column ( P ( μ ( μ μ ( μ μ ( μ μ + q R mq ( μ μ ( μ μ ( μ μ q (C.9 (C.0 C.8 Invere Tranfer Matrix Calculation Ultimately, the tranfer matrix invere can now be exreed comletely by the coefficient calculated above: P (C.
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